Let a1, a2,..., an be a sequence of integers. Show that there exist integers j and k with 1 ≤ j ≤ k ≤ n such that the sum sigma i=j to k ai is a multiple of n

We are tasked with proving that given any sequence of integers \(a_1, a_2, \ldots, a_n\), there exist integers \(j\) and \(k\) such that the sum \( \sum_{i=j}^k a_i \) is a multiple of \(n\).

### Proof Outline:

1. **Prefix Sums**: Define the prefix sums of the sequence:
   \[
   S_m = a_1 + a_2 + \cdots + a_m \quad \text{for } m = 1, 2, \dots, n.
   \]
   Additionally, define \( S_0 = 0 \) for convenience.

2. **R

Let a1, a2,..., an be a sequence of integers. Show that there exist integers j and k with 1 ≤ j ≤ k ≤ n such that the sum sigma i=j to k ai is a multiple of n.

Explore how to prove that for any sequence of integers, there exist integers j and k such that the sum of a subsequence is a multiple of n. The solution uses concepts like prefix sums, modular arithmetic, and the Pigeonhole Principle. Suitable for college-level students.

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